Distant entanglement via photon hopping in a coupled cavity magnomechanical system

We theoretically propose a scheme to generate distant bipartite entanglement between various subsystems in coupled magnomechanical systems where both the microwave cavities are coupled through single photon hopping coupling strength Γ. Each cavity contains a magnon mode and phonon mode and this gives six excitation modes in our model Hamiltonian which are cavity-1 photons, cavity-2 photons, magnon and phonon in cavity-1, and magnon and phonon in cavity-2. We found that significant bipartite entanglement exists between indirectly coupled subsystems in coupled microwave cavities for an appropriate set of parameters regime. Moreover, we also obtain suitable cavity and magnon detuning parameters for a significant distant bipartite entanglement in different bipartitions. In addition, it can be seen that a single photon hopping parameter significantly affects both the degree as well as the transfer of quantum entanglement between various bipartitions. Hence, our present study related to coupled microwave cavity magnomechanical configuration will open new perspectives in coherent control of various quantum correlations including quantum state transfer among macroscopic quantum systems.

Quantum entanglement is a fundamental property of quantum mechanics and has proven to be a key ingredient in various quantum technologies as well as it is an important area of study in both theoretical and experimental quantum physics 1 .In continuous variables (CV) quantum systems which are described by Gaussian states, a very well known mathematical formulation to quantify the amount of bipartite entanglement present is the logarithmic negativity 2 .In the early stages of quantum technology, seminal theoretical and experimental investigations mainly explored only microscopic systems, such as atoms, trapped ions, etc to obtain quantum entanglement 3 .However, the realization of quantum entanglement for various quantum protocols in practical applications and larger-scale quantum technologies often necessitates working at macroscopic level.Major advancements in nanotechnology already provided novel platform such as cavity optomechanical system to study macroscopic bipartite entanglement between a single cavity mode and a vibrating mirror 2 .Subsequently, several studies such as entanglement of two vibrating mirrors [4][5][6][7][8] , Entanglement of multiple cavity modes coupled to single vibrating mirror [9][10][11] , entanglement in Laguerre-Gaussian cavity system [12][13][14][15][16][17][18] explored macroscopic quantum entanglement in cavity optomechanical systems.

The model
The magnomechanical system under consideration consists of two MW cavities connected through single photon hoping factor Ŵ .As shown in Fig. 1, each cavity contain a magnon mode m and a phonon mode b.The magnons in the YIG sphere are considered to be quasiparticles which are integrated by a large-scale collective excitation of spins inside a ferrimagnet, e.g. a YIG sphere 61 .The coupling between the magnon and the MW-field is due to magnetic dipole interaction.In addition, the positioning of YIG sphere in each cavity field is in the zone where there is a maximum magnetic field (See Fig. 1).At the YIG sphere site, the magnetic field of the cavity mode is along the x axis while the drive magnetic field is along the y direction).Furthermore, the bias magnetic field is set in the z direction.In addition, the magnon and phonon modes are coupled to each other via magnetostrictive force, which yields the magnon-phonon coupling 62,63 .The resonance frequencies of the magnon and phonon modes affect the magnetostrictive interaction 24 .In the current study, the mechanical frequency is considered to Figure 1.(a) Graphical representation of the coupled cavity magnomechanical system.Each MV cavity contain a magnon mode in a YIG sphere couples that interact with the cavity mode via magnetic dipole interaction.Furthermore, magnon mode interact with phonon mode via magnetostrictive interaction.The magnetic field of the each cavity modes is set to be in the x-direction, while the drive magnetic field (bias magnetic field) is considered along y-direction (z-direction).(b) The linear coupling diagram of each cavity magnomechanical system is shown.The two cavity modes are coupled via photon hoping Ŵ , while a cavity mode photon c 1 ( c 2 ) is coupled to the magnon mode m 1 ( m 2 ), with coupling strength g 1 ( g 2 ), which then coupled to a phonon mode b 1 ( b 2 ) to with magnomechanical coupling strength g m1 ( g m2 ).
www.nature.com/scientificreports/be much smaller than the magnon frequency, which surely helps to set up the strong dispersive phonon-magnon interaction 61,64 .The Hamiltonian of the coupled magnomechanical system takes the form: where where c k c † k and m k m † k are the annihilation (creation) operator of the the k cavity and magnon mode, respectively.Furthermore, q k and p k are the position and momentum quadratures of the respective mechanical mode of the magnon.In addition ω k , ω m k , and ω b k denote the resonant frequencies of the k-th cavity mode, magnon mode, and mechanical mode.The magnon frequency ω m k can be finely tuned by adjusting the bias magnetic field B through the relation ω m k = γ 0 B , where γ 0 represents the gyromagnetic ratio.Furthermore, the optom- agnonical coupling strength is given by where V represents the Verdet constant of the YIG sphere, ρ spin represents the spin density, n r stands for the refractive index, and V YS = 4πr 3 3 corresponds to the volume of the YIG sphere 65 .We examined the scenario of strong coupling, wherein the interaction between the k-th cavity mode and magnon mode g k surpasses the decay rates of both the magnon and the cavity modes, i.e. g k > κ m k , κ k 25,65,66 .Furthermore, g mk denotes the interaction strength between magnons and phonons, which is generally considered to be quite small.However, it can be improved by employing a MW field to drive the YIG sphere.The Rabi frequency � = ( √ 5/4)γ 0 N spin B 0 67,68 denotes the coupling strength of the drive field with frequency ω 0 and amplitude B 0 = 3.9 × 10 −9 T, where N spin = ρV YS is the total number of spins with the spin density of the YIG ρ spin = 4.22 × 10 27 m −3 and γ 0 = 28 GHz/T.It is also crucial to emphasize that the collective motion of the spins is reduced to bosonic operators m and m † through the Holstein-Primakoff transformation.Additionally, the Rabi frequency is obtained based on the fundamental assumption of having low-lying excitations, specifically when 2Ns ≫ �m † m� , where s = 5 2 represents the spin value of the Fe 3+ ion in the ground state of YIG.Moreover, Ŵ represents the single photon hopping strength between the two cavity mode which is mainly controlled by adjusting the distance between two microwave cavities.However, there are other factor which affect the photon hopping strength like cavity detuning, cavity decay as well as transmission and reflection of the cavity mirrors.
The Hamiltonian of the system can be written as following under the rotating wave approximation at the drive frequency ω 0 : where

Quantum dynamics and entanglement of the coupled magnomechanical system
Because of the interaction between the magnomechanical system and its environment, the system will experience influences from cavity decay, magnon damping, and mechanical damping.By considering these dissipative factors, the system's dynamics can be characterized by a set of quantum Langevin equations : (1) where κ k (κ m k ) represents the decay rate of the k-th cavity (magnon) mode and γ b is the damping rate of the k-th mechanical mode.ξ , m in , and c in k are operators for input noise associated with the k-th mechanical, magnon, and cavity modes, respectively.These noise operators are defined by the following correlation functions 69 : The equilibrium mean number of thermal photons, magnons, and phonons is expressed as Here, k b represents the Boltzmann constant and T denotes the temperature of the environment.
If the magnon mode experiences strong excitation, it implies that |�m�| ≫ 1 .Furthermore, the two microwave cavity fields exhibit large amplitudes due to interactions with the beam splitter interaction between the cavity magnon modes.This allows us to simplify the QLEs by expressing any operator as the sum of its mean value and its fluctuation, i.e. o = �o� + δo [70][71][72] , where (o = p k , q k , c k , m k ) , and then substitute this into Eqs. ( 7), ( 8), (9) and (10).The mean values of the dynamic operators can be calculated as follows: where k , m k = m k0 + g mk q k represents the effective magnon mode detuning which includes the slight shift of frequency due to the magnetostrictive interaction.Now, we introduce the quadrature for the linearised quantum Langevin equations describing fluctuations are: where F (t) and N(t) denote the vectors of quantum fluctuations and input noise, respectively.They are defined as: where and ( 10) www.nature.com/scientificreports/ In addition, the drift matrix M of the present coupled magnomechanical system can be expressed as where G k = i √ 2g mk �m k � governs the effective magnomechanical coupling strength.Furthermore, using Eq. ( 4), By utilizing strong magnon drive, the effective magnomechanical coupling strength can be increased.. Now we investigate the entanglement between various bipartite subsystems with an emphasis on the entanglements of the indirectly coupled mode.According to the Routh-Hurwitz criterion 73 , the system achieves stability only when we obtained negative real parts of all eigenvalues of the drift matrix M , which is verified throughout this manuscript and therefore, the success of our approach hinges on the stability of the proposed system.The system under study is characterized by an 12 × 12 covariance matrix V and its corresponding elements defined as: The following Lyapunov equation can be utilized to determine the covariance matrix of our coupled magnomechanical system 74,75 w here D = di ag[κ 1 (2n 1 + 1), κ 1 (2n 1 + 1), κ 2 (2n 2 + 1), κ 2 (2n 2 + 1), κ m 1 2n m 1 + 1 , κ m 1 (2n m 1 + 1), κ m 2 2n m 2 + 1 , κ m 2 (2n m 2 + 1), 0, γ b (2n b + 1), 0, γ b (2n b + 1)] represents the diffusion matrix, a diagonal matrix that characterizes noise correlations.Furthermore, Eq. ( 24) represents the steady-state correlation matrix.We use the logarithmic negativity to measures the degree of entanglement of the steady state, is given by [75][76][77][78][79] where η − =min eig| 2 j=1 (−σ y ) V 4 | represents the covariance matrix's smallest symplectic eigenvalue.Here, , where V in is a 4 × 4 matrix obtained by extracting the relevant rows and columns from V 4 for the chosen subsystems.The matrix ̺ 1|2 = σ z 1 =diag(1, −1, 1, 1) performs partial transposition on covari- ance matrices.σ 's are the Pauli spin matrices in this context.Moreover, a positive value of logarithmic negativity, given as E N > 0 , highlights the existence of bipartite entanglement between any two given modes in our cavity magnomechanical system.

Results and discussion
As there are six different modes in this coupled cavity magnomechanical system, we investigate into details about the numerical results of different bipartite entanglements.So, we may get bipartite entanglement in any of two modes however the most significant part of our study is to explore the bipartite entanglement present in spatially distant subsystems which we have summarised in Table 1 with symbols.We have used the experimental feasible parameters in our study given as in Table 2.
In Fig. 2, we present five different distant bipartite entanglements as a function of the cavity detunings and gradually changing the hoping factor in our coupled magnomechanical system.For single photon hopping factor Ŵ = 0.5κ c and keeping both the magnon detunings at blue sideband regime i.e. � m 1 = � m 2 = ω b , we obtain the optimal bipartite entanglements E N c 1 −c 2 and E N c 1 −m 2 (E N c 2 −m 1 ) at two different places, however the entanglement Table 1.Adopted notation for the different bipartite subsystem entanglement.

Bipartite subsystem Entanglement symbol Bipartite subsystem Entanglement symbol
Cavity 1-cavity 2 www.nature.com/scientificreports/ ) is concentrated to a specific region of normalized cavity detunings as can be seen in Fig. 2a-c.It can be also seen that the optimal entanglement of these bipartitions exist for different cavity detunings which means that we can shift/transfer entanglement from one bipartition to another through gradually changing both the cavity detunings.Moreover, the bipartite entanglement between two cavity modes E N c 1 −c 2 becomes maximum for � 1 = � 2 = −0.5ωb although even if both the cavity detunings are resonant only with blue sideband regime i.e. � 1 = � 2 = ω b we have significant amount of the bipartite entanglement in E N c 1 −c 2 as shown in Fig. 2a.In Fig. 2b we study the bipartite entanglement E N c 1 −m 2 (E N c 2 −m 1 ) which attains maximum value either when both the cavity detunings are resonant with the driving field, i.e. 1 = 2 = 0 or are resonant with red sideband regime, i.e. � 1 = � 2 = −ω b .Moreover when both the cavity detunings are kept in resonance (symetric case) with red sideband regime, the bipartite entanglement E N c 1 −b 2 (E N c 2 −b 1 ) attains its maximum value as shown in Fig. 2c.Furthermore, it can be seen that if cavity detunigs for both the cavities are kept fixed and in resonance with blue sideband regime i.e. � 1 = � 2 = ω b then all the above mentioned bipartite entanglements have significant values on gradually varying � m 1 /ω b from 0.7 to 1.1 as shown in Fig. 2d-f, however, the entanglement can be transferred among these bipartitions by altering the � m 2 /ω b .So, in this case to get significant amount of bipartite  2. entanglements between different modes either both the cavity detunings or both the magnon detunings should be kept at blue sideband regime of the phonons due to deformation of YIG sphere.This is because it leads to anti-stokes process which results in significant cooling of phonons and enhance the bipartite entanglement between the different bipartions.
Next, we plot five different distant bipartite entanglements as a function of � 1 /ω b and � 2 /ω b for different single photon hopping factor Ŵ , while keeping both the magnon detuning in resonance with blue sideband regime i.e. � m 1 = � m 2 = ω b in Fig. 3.For Ŵ = 0.5ω b , the quantity E N c 1 −c 2 attains optimal value around i.e. � 1 = � 2 = −0.5ωb whereas for off resonant cavities, we also get finite values of E N c 1 −c 2 as shown in Fig. 3a.However, the cavity-magnon entanglements E N c 1 −m 2 (= E N c 2 −m 1 ) become maximum for two different values of cavity detunings which are 1 = 2 = 0 and � 1 = � 2 = −ω b as shown in Fig. 3b.In addition, we obtained the optimal cavity-phonon entanglements −ω b as shown in Fig. 3c.It can be seen that if we increase the single photon hopping factor upto Ŵ = 0.8ω b then the bipartite entanglement in between both the cavity modes E N c 1 −c 2 becomes maximum for two cases i.e. either both the cavity detunings should be in red sideband regime ( � 1 = � 2 = −ω b ) or in blue sideband regime ( � 1 = � 2 = ω b ) as shown in Fig. 3d.In addition, in the density plots of the bipartite quantities E N c 1 −m 2 = E N c 2 −m 1 the region corresponding to red sideband regime start to decrease whereas the region corresponding to resonant cavities increases as shown in Fig. 3e.Moreover, the quantities show the finite values for a broad range of cavity detunings and attain maximum value for � 1 = � 2 = −1.5ωb as shown in Fig. 3f.On further increasing the value of Ŵ and keeping it at Ŵ = ω b , the quantity E N c 1 −c 2 again becomes maximum for two cases i.e. for � 1 = � 2 = −0.5ωb and � 1 = � 2 = −1.5ωb as shown in Fig. 3g whereas the quantities E N c 1 −m 2 = E N c 2 −m 1 attain maximum value only when both the cavity detunings are nearly resonant with blue sideband regime as given in Fig. 3h.However, both the quantities b 1 attain maximum value only for very far off-resonant cavities � 1 = � 2 = −2ω b whereas for a broad range of negative cavity detunings both these distant entanglements almost become negligible however for a positive value of � 1 /ω b and � 2 /ω b both the bipartite entanglements attain finite values as shown in Fig. 3i.Overall, it can be also seen from Fig. 3 that as the cavity-cavity photon hopping strength Ŵ continue to increase, concentration of various bipartite entanglements in density plots decrease significantly.This is because within a certain range, the photon hopping strength is positively correlated with the bipartite entanglement, but when it is increases continuously, the quantum system will undergo degradation, leading to a decrease in bipartite entanglement.Now, we study the effects of varying photon hopping factor Ŵ/κ c and normalised first cavity detuning � 1 /ω b on these five bipartite entanglements.It is important to mention here that we have taken two cases: symmetric case ( � 1 = � 2 = ω b ), which correspond to upper panel and non symmetric case ( � 1 = −� 2 = ω b ) corresponding to lower panel while keeping second cavity detuning � 2 /ω b fixed in Fig. 4. It can be seen that for ) become almost zero for positive values of �/ω b as shown in Fig. 5a.It can be seen that for this value of Ŵ a significant amount of entanglement transfer takes place from ) and ) at �/ω b ≈ -0.3 and -1.2.For Ŵ = 0.8ω b , the bipartite entanglement E N c 1 −c 2 become finite for �/ω b varying in the range of (0.3)-(1.3) and (-0.5)-(-1.3)as shown in Fig. 5b.It can be also seen that the bipartite quantities E N m 2 −b 2 (E N m 1 −b 1 ) almost get around 0.2 for positive as well as negative values of �/ω b except for certain values of �/ω b ≈ 0.2 and −1.5 whereas ) has finite values upto �/ω b ≈ 0. 5 and E ) becomes zero even for negative values of �/ω b .In this case we get maximum entanglement transfer from ) and ) around �/ω b ≈ 0.1 and -1.5.If we increase further single photon hopping factor upto Ŵ = ω b then the quantity E N c 1 −c 2 remains ) qualitatively remains the same as depicted in Fig. 5c.It can be seen that the maximum entanglement transfer from ) and ) takes place around values �/ω b ≈ 0.5 and -1.8.Now for antisymmetric cavities � = � 1 = −� 2 = ω b and single photon hopping factor Ŵ = 0.5ω b it can be seen that both the bipartite entanglements E N m 2 −b 2 (E N m 1 −b 1 ) have finite values with a varying �/ω b although for few values both become zero as shown in Fig. 5d.All other bipartite entanglements have very small values for this value of Ŵ .For Ŵ = 0.8ω b the bipartite entanglement E N c 1 −c 2 becomes zero whereas the quantities ) have finite values from (0.1) − (0.25) as shown in Fig. 5d.Moreover, the bipartite entanglements ) increases for this value of Ŵ and become finite for a varying �/ω b in between the range of (-1)-( 1) whereas ) also increases and varies from 0-0.07(0.08)with �/ω b as depicted in Fig. 5d.With a further increment in Ŵ both the bipartite entanglements ) becomes finite over whole range of varying �/ω b whereas all other bipartite entanglements qualitatively remain the same (like earlier case of Ŵ = 0.8ω b ) as shown in Fig. 5f.It can be also seen that in case of symmetric microwave cavities when both the cavity detunings are kept fixed at blue sideband regime and only anti stokes process dominates, we have finite cavity-cavity entanglement whereas for antisymmetric cavities we get both stokes and anti stokes processes and hence it almost gives negligible cavity-cavity entanglement for any value of single photon hopping strength Ŵ .In addition, for symmetric cavities we also get maximum entanglement transfer from directly coupled modes (magnon-phonon) to indirectly coupled modes (cavity-magnon) as shown in Fig. 5.
We study the density plots of different distant bipartite entanglements as a function of environmental temperature T and single photon hopping factor Ŵ/κ c for symmetric case i.e. � 1 = � 2 = ω b (upper panel) and non symmetric case i.e. � 1 = −� 2 = ω b (lower panel).For second cavity detuning � 2 = ω b it can be seen that for T ∼ 0.1K − 0.15K , all the three bipartite entanglements E N c 1 −c 2 , E N c 1 −m 2 and E N c 2 −m 1 are finite for a very narrow range of Ŵ/κ c as shown in Fig. 6a-c

Conclusion
We present an experimentally feasible scheme based on coupled magnomechanical system where two microwave cavities are coupled through single photon hopping parameter Ŵ and each cavity also contains a magnon mode and phonon mode.We have investigated continuous variable entanglement between distant bipartitions for an appropriate set of both cavities and magnons detuning and their decay rates.Hence, it can be seen that bipartite entanglement between indirectly coupled systems are substantial in our proposed scheme.Moreover, in our present scheme cavity-cavity coupling strength also plays a key role in the degree of bipartite entanglement and its transfer among different direct and indirect modes.This scheme may prove to be significant for processing continuous variable quantum information in quantum memory protocols. https://doi.org/10.1038/s41598-023-48825-8

Figure 4 .
Figure 4. Density plot of bipartite entanglement in (a,f)E N c 1 −c 2 ; in (b,g) E N c 1 −m 2 ; in (c,h) E N c 2 −m 1 ; in (d,i) E N c 1 −b 2and (e,j) E N c 2 −b 1 versus � 1 /ω b and Ŵ/ω b for � 2 = ω b in (a-e) and for � 2 = −ω b in (f-j).The other parameters are same as in Fig.3.